The Stacks project

Proof. We have already seen that (1) and (2) imply that $R$ is Noetherian, see Lemma 10.164.1. Let $\mathfrak p \subset R$ be a prime ideal. Choose a prime $\mathfrak q \subset S$ lying over $\mathfrak p$ which corresponds to a minimal prime of the fibre ring $S \otimes _ R \kappa (\mathfrak p)$. Then $A = R_{\mathfrak p} \to S_{\mathfrak q} = B$ is a flat local ring homomorphism of Noetherian local rings with $\mathfrak m_ AB$ an ideal of definition of $B$. Hence $\dim (A) = \dim (B)$ (Lemma 10.112.7) and $\text{depth}(A) = \text{depth}(B)$ (Lemma 10.163.2). Hence since $B$ has $(S_ k)$ we see that $A$ has $(S_ k)$. $\square$